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Journal of Integer Sequences, Vol. 20 (2017),
Article 17.4.4
23 11
Some Formulae for Products of Geometric
Polynomials with Applications
Levent Kargın
Akseki Vocational School
Alanya Alaaddin Keykubat University
TR-07630 Antalya
Turkey
[email protected]
Abstract
In this paper, we evaluate sums and integrals of products of two geometric polynomials and obtain new explicit formulas for geometric polynomials and numbers. As
a consequence of these results, we give new explicit formulas for p-Bernoulli numbers,
Apostol-Bernoulli functions, and integrals of products of two Apostol-Bernoulli functions.
1
Introduction
Let nk be the Stirling numbers of the second kind [16]. Geometric polynomials are defined
by [33]
n X
n
k!y k .
(1)
wn (y) =
k
k=0
They have the exponential generating function
∞
X
1
tn
=
,
w
(y)
n
1 − y (et − 1) n=0
n!
1
(2)
and are related to the geometric series by [5]
m
∞
X
d
1
1
y
m k
y
, |y| < 1.
=
wm
k y =
dy
1−y
1−y
1−y
k=0
In addition, the following recurrence relation holds for the geometric polynomials [13]:
wn+1 (y) = y
d
wn (y) + ywn (y) .
dy
(3)
The nth geometric number (ordered Bell number, or Fubini number, or preferred arrangement number) [12, 17, 33], wn , is defined by
n X
n
k!,
wn (1) = wn =
k
k=0
(4)
and counts all the possible set partitions of an n element set such that the order of the blocks
matters. Besides with this combinatorial property, these numbers are seen in the evaluation
of the following series
∞
X
kn
= 2wn .
(5)
2k
k=0
In the literature, numerous identities concerned with these polynomials and numbers were
obtained [5, 6, 7, 8, 14, 28] and some generalizations were given [5, 15, 20, 29].
The sums of products of various polynomials and numbers with or without binomial
coefficients have been studied [2, 10, 21, 25, 26, 32, 34]. One of the famous results is [19, Eq.
50.11.2]
n X
n
Bk (x) Bn−k (y) = (1 − n) Bn (x + y − 1) + (x + y − 1) nBn−1 (x + y − 1) ,
k
k=0
(6)
where Bn (x) is the nth Bernoulli polynomial. In addition, the integrals of products of
various polynomials and functions have been studied [3, 6, 11, 24, 27]. For example, the
following interesting integral for a product of two Bernoulli polynomials appears in the book
by Nörlund [30, p. 31]: For all k + m ≥ 2,
Z1
Bk (x) Bm (x) dx =
k!m!
Bk+m .
(k + m)!
0
Here, Bn is the nth Bernoulli number defined by Bn = Bn (0).
2
(7)
The main purpose of this paper is to obtain analogues of (6) and (7) for geometric
polynomials, which generalize the binomial formulas [13]
n X
n
k
k=0
2
n X
n
k=0
k
wk = 2wn , n > 0,
(−1)k wk = (−1)n wn + 1, n ≥ 0,
(8)
(9)
and the integral [22]
Z
0
wn (y) dy = Bn , n > 0.
(10)
−1
The paper is outlined as follows. We derive sums of products of two geometric polynomials
and explicit formulas for these polynomials and numbers in Section 2. In Section 3, we give
integrals of products of two geometric polynomials. As an application of this result, an
explicit formula for p-Bernoulli numbers is obtained. Finally, in Section 4, we give a new
explicit formula and integrals of products of Apostol-Bernoulli functions.
2
Sums of products of geometric polynomials
In this section, we define two variable geometric polynomials and obtain some basic properties
which give us new formulas for wn (y). Moreover, we consider the sums of products of two
geometric polynomials.
Two variable geometric polynomials are defined by means of the following generating
function:
∞
X
ext
tn
=
.
(11)
wn (x; y)
t − 1)
n!
1
−
y
(e
n=0
As some special cases of (11), we have
wn (0; y) = wn (y) and wn (0; 1) = wn .
We can rewrite (11) as
∞
X
n=0
wn (x; y)
1
tn
=
ext
n!
1 − y (et − 1)
∞
X
∞
tn X n tn
=
wn (y)
x
n! n=0 n!
n=0
∞ X
n X
tn
n
=
.
wk (y) xn−k
k
n!
n=0 k=0
3
(12)
Comparing the coefficients of
tn
n!
yields
wn (x; y) =
n X
n
k
k=0
wk (y) xn−k .
(13)
Besides, from (11) we have
∞
X
wn (x + 1; y) − wn (x; y)
n=0
tn
ext (et − 1)
=
n!
1 − y (et − 1)
1
ext
− ext
t
y 1 − y (e − 1)
∞
tn
1X
.
wn (x; y) − xn
=
y n=0
n!
=
Comparing the coefficients of
tn
n!
gives
ywn (x + 1; y) = (1 + y) wn (x; y) − xn .
(14)
Thus, setting x = 0 and x = −1 in (14), we find
ywn (1; y) = (1 + y) wn (y) , n > 0,
(1 + y) wn (−1; y) = ywn (y) + (−1)n , n ≥ 0,
(15)
(16)
respectively. Combining these relations with (13) gives equations (8) and (9) which were
obtained by using Euler-Seidel matrix method in [13].
Now, we want to give the generalization of the binomial formula (8). Derivative of (11)
can be written as
∂
xext
ext
yet
ext
=
+
.
∂t 1 − y (et − 1)
1 − y (et − 1) 1 − y (et − 1) 1 − y (et − 1)
Taking x = x1 + x2 − 1 leads to
X
∞
∂
tn
ext
=
w
(x
+
x
−
1;
y)
,
n+1
1
2
∂t 1 − y (et − 1)
n!
n=0
and
∞
X
xext
tn
=
(x
+
x
−
1)
,
w
(x
+
x
−
1;
y)
1
2
n
1
2
1 − y (et − 1)
n!
n=0
! ∞
!
∞
X
X
ext
tn
tn
yet
wn (x1 ; y)
wn (x2 ; y)
=y
1 − y (et − 1) 1 − y (et − 1)
n!
n!
n=0
n=0
∞ X
n X
tn
n
wk (x1 ; y) wn−k (x2 ; y) .
=y
k
n!
n=0 k=0
4
By equating the coefficients of
y
n X
n
k=0
k
tn
,
n!
we get
wk (x1 ; y) wn−k (x2 ; y) = wn+1 (x1 + x2 − 1; y) − (x1 + x2 − 1) wn (x1 + x2 − 1; y) .
For x1 = x2 = 0 in the above equation, by using (16) gives the sums of products of the
geometric polynomials.
Theorem 1. For n ≥ 0,
(y + 1)
n X
n
k=0
When y = 1 this becomes
k
wk (y) wn−k (y) = wn+1 (y) + wn (y) .
n X
n
wk wn−k = wn+1 + wn .
2
k
k=0
(17)
(18)
We note that Boyadzhiev and Dil [9, Proposition 9] proved a more general form of
Theorem 1 by different way.
Now, we investigate the sums of products of the geometric polynomials for different y
values in the following theorem.
Theorem 2. For n ≥ 0 and y1 6= y2 ,
n X
y2 wn (y2 ) − y1 wn (y1 )
n
wk (y1 ) wn−k (y2 ) =
.
k
y2 − y1
k=0
(19)
Proof. We write
e x1 t
e x2 t
(1 − y1 (et − 1)) (1 − y2 (et − 1))
y2
e(x1 +x2 )t
e(x1 +x2 )t
y1
=
−
.
y2 − y1 1 − y2 (et − 1) y2 − y1 1 − y1 (et − 1)
Using the same method as in the proof of Theorem 1, we have
n X
n
k=0
k
wk (x1 ; y1 ) wn−k (x2 ; y2 ) =
y2 wn (x1 + x2 ; y2 ) − y1 wn (x1 + x2 ; y1 )
.
y2 − y1
Setting x1 = x2 = 0 in the above equation gives the desired equation.
5
(20)
As we know, for y = 1, geometric polynomials reduce to geometric numbers. We now
point out (see (24)) that geometric numbers also arise for other values of y. If we take y − 1
in place of y in (11), we have
wn (x; y − 1) = (−1)n wn (1 − x; −y) .
(21)
Setting x = 0 in the above equation and using the relation (15), we have a reflection formula
y
wn (−y − 1) , n > 0.
(22)
y+1
Therefore, employing (1) gives a new explicit formula for geometric polynomials as in the
following theorem.
wn (y) = (−1)n
Theorem 3. For n > 0, we obtain
wn (y) = y
n X
n
k=1
k
(−1)n+k k! (y + 1)k−1 .
(23)
Note that, when y = 1 (23) reduces to [18, Theorem 4.2]. Moreover, from (22), we get
two conclusions
−1
w2k
= 0 and wn (−2) = (−1)n 2wn .
(24)
2
The first part of (24) was given in [9, Corollary 17]. If we take y1 = −2 and y2 = 1 in (19)
and use the second part of (24), we obtain the alternating sums of products of geometric
numbers.
Corollary 4. For n > 0, we have
(
n X
0,
n
(−1)k wk wn−k = 4
k
w ,
3 n
k=0
if n is odd;
if n is even.
Finally, we obtain a new explicit formula for geometric polynomials and numbers in the
following theorem.
Theorem 5. For y 6=
and n ≥ 0,
n n+1
X
(y + 1) y k + (−1)k+1
n
k 2
k!y
wn (y) =
.
k
(2y + 1)k+1
k=0
(25)
n X
2n+2 + (−1)k+1
n
k!
wn =
, n ≥ 0.
k
3k+1
k=0
(26)
−1
2
When y = 1 this becomes
When y = −2 this becomes
n
X
k−1 n+k+1
2
2
+
1
n
k!
, n > 0.
wn =
(−1)n−k
k+1
3
k
k=0
6
(27)
Proof. If we take
1
y 2 −1
in place of y in (2), we arrive at
1
y2 − 1
1
1 =
.
+
2y 2 y − et y + et
1 − y21−1 (e2t − 1)
Each of the functions in the above equation can be written as
n
∞
X
t
1
1
n
=
2
w
,
n
1
2
y − 1 n!
1 − y2 −1 (e2t − 1) n=0
n
∞
1
t
y X
1
=
,
wn
t
y−e
y − 1 n=0
y − 1 n!
n
∞
1
y X
t
−1
=
.
wn
t
y+e
y + 1 n=0
y + 1 n!
By equating the coefficients of
wn (y) = 2
tn
,
n!
n+1
(28)
(29)
(30)
(31)
we have
(1 + y) wn
y2
1 + 2y
− (1 + 2y) wn (−y) .
(32)
Finally, using (1) in the right hand side of the above equation yields (25).
3
Integrals of products of geometric polynomials
In this section, we deal with an integral for a product of two geometric polynomials. First
we need following lemmas.
Lemma 6. For all k ≥ 0 and n ≥ 1, we have
Z0
−1
where
n k
k (−1)k X k + 1
y wn (y) dy =
Bn+j ,
k! j=0 j + 1
k
(33)
is the Stirling number of the fist kind [16].
Proof. We prove (33) by induction on k. The case k = 0 of (33) is known from (10). If we
integrate both sides of (3) with respect to y from −1 to 0 and apply integration by parts,
we have
Z0
Z0
d
wn+1 (y) dy = y
wn (y) + ywn (y) dy
dy
−1
−1
= [y (wn (y) + ywn (y))]0−1 −
Z0
−1
7
wn (y) + ywn (y) dy.
So, using (10) yields the case k = 1 of (33) as
Z0
ywn (y) dy = − (Bn+1 + Bn ) .
−1
Multiplying both sides of (3) by y and integrating it with respect to y from −1 to 0, we
obtain
Z0
Z0
d
ywn+1 (y) dy = y 2
wn (y) + ywn (y) dy.
dy
−1
−1
Applying integration by parts and using (10) yield the case k = 2 of (33) as
2
Z0
y 2 wn (y) dy = Bn+2 + 3Bn+1 + 2Bn .
−1
If we multiply both sides of (3) by y k and integrating it with respect to y from −1 to 0, we
obtain
Z0
Z0
d
wn (y) + ywn (y) dy.
y k wn+1 (y) dy = y k+1
dy
−1
−1
Applying integration by parts to the right hand side of the above equation and considering
Z0
−1
k (−1)k X k + 1
y wn (y) dy =
Bn+j ,
k! j=0 j + 1
k
we have
Z0
−1
y
k+1
k k
(−1)k+1 X k + 1
(−1)k+1 X
k+1
Fn+1 (y) dy =
Bn+j+1 +
Bn+j .
(k + 1)
(k + 1)! j=0 j + 1
(k + 1)! j=0
j+1
Finally, the well-known relations
n+1
n
n
n
=n
+
and
= (n − 1)!,
k
k
k−1
1
give that the statement is true for k + 1.
Lemma 7. For any non-negative integer m and j,
m X
m k+1
k
m m
.
(−1) = (−1)
j
j+1
k
k=j
8
(34)
Proof. We rewrite this equation into matrix form by using the matrices
i
i
i+j i + 1
.
, (B)i,j =
, (S2 )i,j =
(S1 )i,j = (−1)
j
j
j+1
These can be considered as infinite matrices so that the statement we are going to prove
takes the form
S2 S1 = B −1 ,
i+k i
. The above equation
where the element-wise inverse of the matrix B is (B)−1
i,k = (−1)
k
is equivalent to
S1 = B −1 S2−1 = (S2 B)−1 .
The matrix on the right hand side is easily decipherable. Element-wise, it is
−1
((S2 B) )i,j =
i X
k
i
k
j
i+1
j+1
k=0
.
The latter sum simply equals to
i X
i
k
k=0
k
j
=
[16, Eq. 6.15]. Hence, our original statement equals to the matrix equation
i+1
−1
.
(S1 )i,j =
j+1
This is nothing but the reformulation of the fact that the second and signed first kind Stirling
matrices are inverses of each other.
Now, we are ready to give the integrals of products of geometric polynomials. Using (1),
we have
Z0 X
Z0
m m
k!y k wn (y) dy.
wm (y) wn (y) dy =
k
k=0
−1
−1
Then, interchanging the sum and integral in the above equation and using (33) yield
Z0
−1
wm (y) wn (y) dy =
m X
m X
m k+1
j=0 k=j
k
Finally, using Lemma 7 gives the following theorem.
9
j+1
(−1)k Bn+j .
Theorem 8. For all m ≥ 0 and n ≥ 1, we have
Z0
wm (y) wn (y) dy = (−1)
m
m X
m
j
j=0
−1
(35)
Bn+j .
Using the representation (1) in (35) and integrating termwise, one obtains
n X
m X
m (−1)k+j k!j!
n
k+j+1
j
k
k=0 j=0
= (−1)
m
m X
m
j=0
j
Bn+j .
This double sum identity extends to the explicit formula [16, p. 560]
Bn =
n X
n
k
k=0
(−1)k
k!
.
k+1
In order to give an application of Lemma 6, we now emphasize the summation in the
right hand of (33). Rahmani [31] defined p-Bernoulli numbers as
∞
X
Bn,p
n=0
tn
=
n!
2 F1
1, 1; p + 2; 1 − et ,
where 2 F1 (a, b; c; z) denotes the Gaussian hypergeometric function [1]. These numbers can
be written in terms of Stirling numbers of the first kind as follows:
p
X
p!
p
Bn,p , n, p ≥ 0.
(−1)
Bn+j =
p+1
j
j=0
j
From the above equation, we have
p
X
(−1)
j=0
j+1
(p + 1)!
p+1
Bn+j =
Bn−1,p+1 , n ≥ 1, p ≥ 0.
j+1
p+2
Moreover, when n is odd or even, we have
(−1)j+1 Bn+j = Bn+j or (−1)j+1 Bn+j = −Bn+j , n > 1,
respectively. Therefore, we obtain
p X
p+1
j=0
j+1
Bn+j =
(
(p+1)!
Bn−1,p+1 ,
p+2
(p+1)!
− p+2 Bn−1,p+1 ,
10
if n is odd;
if n is even.
(36)
Using the above equation, (33) can be written as
(
Z0
p+1
Bn−1,p+1 ,
if n is odd;
(−1)p p+2
y p wn (y) dy =
p+1 p+1
(−1)
B
, if n is even,
p+2 n−1,p+1
−1
where n > 1, p ≥ 0. On the other hand, using (1) in the left part of (3), a new explicit
formula for p-Bernoulli numbers is obtained.
Theorem 9. For n > 1 and p > 0,
B2n−1,p
and
B2n,p
4
2n−1 p + 1 X 2n − 1 (−1)k+1 (k + 1)!
=
p k=0 k + 1
k+p+1
2n p + 1 X 2n + 1 (−1)k (k + 1)!
=
.
p k=0 k + 1
k+p+1
Applications
Apostol-Bernoulli functions Bn (λ) have the following explicit expression:
k
n−1 n X n−1
λ
Bn (λ) =
k!
, λ ∈ C\{1}.
k
λ − 1 k=0
1−λ
(37)
Thus, for λ 6= 1,
B0 (λ) = 0, B1 (λ) =
−2λ
1
, ...
, B2 (λ) =
λ−1
(λ − 1)2
The functions Bn (λ) are rational functions in the variable λ. Apostol [4] introduced these
functions in order to evaluate the Lerch transcendent (also known as the Lerch zeta function)
for negative integer values of s. Also, these functions were studied and generalized recently
in a number of papers, under the name Apostol-Bernoulli numbers.
Comparing (37) and (1), Boyadzhiev [7] showed that Apostol-Bernoulli functions can be
expressed by geometric polynomials as
n+1
λ
Bn+1 (λ) =
, λ ∈ C\{1}.
(38)
wn
λ−1
1−λ
We can use this relation to obtain some new properties of Bn (λ). For example, setting
−λ
y = λ−1
in (23), we have [18, Theorem 4.3]
k+1
n X
n
1
Bn+1 (λ)
n
k!
, λ 6= 1, n ≥ 0.
= (−1) λ
(n + 1)
λ−1
k
k=0
11
Similarly, from Theorem 1, we get the sums of products of Apostol-Bernoulli functions as
given in [23, Corollary 1.3]. Moreover, using equation (25) of Theorem 5 gives a new explicit
formula for Apostol-Bernoulli functions.
Corollary 10. For λ 6= ±1 and n ≥ 0,
n (−λ)k 2n+1 λk + (λ − 1)k+1
Bn+1 (λ) X n
k!
=
.
2 − 1)k+1
k
(n + 1)
(λ
k=0
To give a different application of the relation (38), we first deal with Lemma 6. Replacing
λ
in (33), we have
y with 1−λ
Z0
−∞
k n+1X k+1
Bn+j ,
Bn+1 (λ) dλ =
k! j=0 j + 1
(λ − 1)k+1
λk
where k ≥ 0 and n ≥ 1. Then, from Theorem 8, we obtain the integrals of products of
Apostol-Bernoulli functions as given in the following corollary.
Corollary 11. For all m ≥ 0 and n ≥ 1, we have
Z0
m
Bm (λ) Bn (λ) dλ = (−1) (m + 1) (n + 1)
j=0
−∞
5
m X
m
j
Bn+j .
Acknowledgments
The author would like to thank to Professor Istvan Mező for his valuable suggestions regarding improvement of the paper.
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2010 Mathematics Subject Classification: Primary 11B83; Secondary 11B68, 11B75.
Keywords: geometric number, geometric polynomial, Apostol-Bernoulli function, p-Bernoulli
number.
(Concerned with sequence A000670.)
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Received September 21 2016; revised version received February 2 2017; February 6 2017.
Published in Journal of Integer Sequences, February 9 2017.
Return to Journal of Integer Sequences home page.
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